We introduce a unifying framework for the construction of holographic tensor
networks, based on the theory of hyperbolic buildings. The underlying dualities
relate a bulk space to a boundary which can be homeomorphic to a sphere, but
also to more general spaces like a Menger sponge type fractal. In this general
setting, we give a precise construction of a large family of bulk regions that
satisfy complementary recovery. For these regions, our networks obey a
Ryu–Takayanagi formula. The areas of Ryu–Takayanagi surfaces are controlled
by the Hausdorff dimension of the boundary, and consistently generalize the
behavior of holographic entanglement entropy in integer dimensions to the
non-integer case. Our construction recovers HaPPY–like codes in all
dimensions, and generalizes the geometry of Bruhat–Tits trees. It also
provides examples of infinite-dimensional nets of holographic conditional
expectations, and opens a path towards the study of conformal field theory and
holography on fractal spaces.
